Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x-7y &= 9 \\ 6x+9y &= -9\end{align*}$
Answer: Begin by moving the $y$ -term in the second equation to the right side of the equation. $6x = -9y-9$ Divide both sides by $6$ to isolate $x$ $x = {-\dfrac{3}{2}y - \dfrac{3}{2}}$ Substitute this expression for $x$ in the first equation. $-5({-\dfrac{3}{2}y - \dfrac{3}{2}}) - 7y = 9$ $\dfrac{15}{2}y + \dfrac{15}{2} - 7y = 9$ Simplify by combining terms, then solve for $y$ $\dfrac{1}{2}y + \dfrac{15}{2} = 9$ $\dfrac{1}{2}y = \dfrac{3}{2}$ $y = 3$ Substitute $3$ for $y$ in the top equation. $-5x-7( 3) = 9$ $-5x-21 = 9$ $-5x = 30$ $x = -6$ The solution is $\enspace x = -6, \enspace y = 3$.